Question

In the closest packing of atoms: (a) the size of tetrahedral void is greater than that of octahedral void (b) the size of tetrahedral void is smaller than that of octahedral void (c) the size of tetrahedral void is equal to that of octahedral void (d) the size of tetrahedral void may be greater or smaller or equal to that of octahedral void depending upon the size of atoms

Short answer

In the closest packing of atoms, the size of the tetrahedral void is smaller than that of the octahedral void.

Step-by-step solution

Understanding closest packing of atoms

In the closest packing of atoms, each atom is in contact with a number of other atoms. There are interspaces or voids where other atoms can fit in. There are two main types of voids: tetrahedral voids and octahedral voids. Tetrahedral voids are formed by four atoms located at the corners of a tetrahedron. Octahedral voids are formed by six atoms at the corners of an octahedron.

Comparing the size of tetrahedral and octahedral voids

The sizes of these voids are determined by the arrangement of atoms around them. A tetrahedral void is smaller as it is surrounded by four atoms, while an octahedral void is larger as it is surrounded by six atoms. The ratio of the radii of the tetrahedral void (r_t) to the octahedral void (r_o) can be calculated from the geometry of closest packing and is found to be r_t:r_o = 0.225:0.414, which means the tetrahedral void is smaller than the octahedral void.

Conclusion

The size of the tetrahedral void is always smaller than that of the octahedral void in closest packing of atoms, irrespective of the size of the atoms involved. Hence, option (b) the size of tetrahedral void is smaller than that of octahedral void, is the correct answer.

Key concepts

Tetrahedral and Octahedral Voids

Imagine looking into a crowded room from above, where people are standing as close to each other as possible. In a similar fashion, atoms pack closely together in a solid, leaving behind small 'gaps' or 'voids' between them. These voids are critical in determining many properties of the material.

In closest packing, two primary types of voids are formed: tetrahedral and octahedral. A tetrahedral void is like a pyramid with a triangular base, with atoms at its four corners. It's nestled among atoms and is relatively small. On the other hand, an octahedral void forms when six atoms are positioned at the corners of an octahedron, resembling two back-to-back pyramids with a square base. This void is larger due to its arrangement.

Having fewer atoms around, tetrahedral voids are smaller in size. This makes sense, considering the geometric constraints when we arrange spheres closely together – the space in the center of four touching spheres is smaller than that in the center of six. A good way to visualize this is using models or drawings, where the arrangement and the voids become clear. In solids, these voids can be occupied by smaller atoms, with tetrahedral voids often being the site for smaller ion placement in ionic compounds.

Atomic Arrangement in Solids

Atoms in solids strive for stability and economy of space, which leads them to adopt the tightest packing arrangements. There are several ways atoms can pack in a solid, but the goal is to maximize the atoms' contact with each other while minimizing the space between them.

Closest packing in solids can be of two types: hexagonal closest packing (hcp) and cubic closest packing (ccp), also known as face-centered cubic (fcc). In both types, each atom is surrounded by 12 others, creating a highly efficient and dense structure. These packings lead to the aforementioned tetrahedral and octahedral voids.

The efficiency of atomic packing not only affects the density and void distribution in a solid but also influences its thermal, electrical properties, and overall stability. The arrangement can be related to simpler geometries such as stacking layers of spheres in a way that each lies in the crevice of the layer beneath - akin to how oranges are stacked in a crate. This basic principle extends to the atomic level, giving rise to complex crystalline structures.

Geometry of Closest Packing

At the heart of materials science is the geometry of closest packing – a beautiful and infinitely practical pattern dictating how spheres, or in our case, atoms, fit together in the tightest possible configuration. The geometry of closest packing describes how atoms, imagined as uniform spheres, can be arranged to occupy the least volume.

In a closest packed structure, each atom touches several others, leading to a highly organized arrangement. From a geometric viewpoint, this leads to the aforementioned two primary structures – hcp and ccp (fcc). These structures can be envisioned as layers of atoms. In hcp, the layers are stacked directly over each other, while in ccp, the third layer is shifted relative to the first, repeating every third layer. To illustrate, think of stacking oranges or cannonballs in a pyramid; each orange rests in the dip created by the three below it.

Ratio of Voids in Closest Packing

In a ccp or fcc arrangement, the ratio of the total volume of the spheres to the volume of the cell is approximately 74%, which is the maximum possible packing efficiency for spheres. This leaves about 26% of the space as void, which, as shown in the exercise, includes both tetrahedral and octahedral voids.

The understanding of this geometry is not just an academic exercise. It's foundational to creating new materials with desired properties, predicting the behavior of compounds under various conditions, and even in decoding the mysteries of naturally occurring mineral formations.